FIG. 16 is an explanatory drawing showing measurement of the position of a target which is carried out by a positioning and tracking device disclosed in the following nonpatent reference 1.
In the example shown in FIG. 16, the position of the target, such as an airplane or a satellite, is measured by using distance differences provided by four receiving stations, and DOPs (Dilutions Of Precision) which are evaluation indices for the positioning accuracy for the target are calculated from the measured position of the target.
Hereafter, a method of calculating evaluation indices DOPs for the positioning accuracy for a target will be explained.
In the case of measuring the three-dimensional position of a target by using distance differences provided by receiving stations, the required minimum number of receiving stations is four. Although four or more receiving stations can be used when measuring the three-dimensional position of a target, the three-dimensional position cannot be calculated in the case of using three receiving stations.
Further, in the case of measuring the two-dimensional position of a target by using distance differences provided by receiving stations, the required minimum number of receiving stations is three. Although three or more receiving stations can be used when measuring the two-dimensional position of a target, the two-dimensional position cannot be calculated in the case of using two receiving stations.
When the target is an airplane (including a satellite or the like), the distance between the target and a receiving station can be measured by the receiving station receiving a signal transmitted from the transponder of the target.
Assuming that the distance measured by a receiving station (i) (i=1, 2, 3) is expressed by ri and a receiving station (0) is a reference receiving station (referred to as the “reference station” from here on), the distance difference between the receiving station (i) and the reference station is given by ri−r0. In this specification, the distance difference between the receiving station (i) and the reference station is referred to as the distance difference of the receiving station (i) in some cases.
When the distance difference between each of the receiving stations (1) to (3) and the reference station in FIG. 16 is expressed by mathematical expressions, these expressions are given by the following equations (1) to (3). In this specification, a symbol to which an underline is attached shows a vector.f1(Ztgt)=r1−r0=|Ztgt−Zsns1|−|Ztgt−Zsns0|  (1)f2(Ztgt)=r2−r0=|Ztgt−Zsns2|−Ztgt−Zsns0|  (2)f3(Ztgt)=r3−r0=|Ztgt−Zsns3|−|Ztgt−Zsns0|(3)r0=|Ztgt−Zsns0|(4)r1=|Ztgt−Zsns1|  (5)r2=|Ztgt−Zsns2|  (6)r3=|Ztgt−Zsns3|  (7)Ztgt=[X,Y,Z]′  (8)Zsnsi=[Xsnsi/Ysnsi/Zsnsi]′  (9)
In the equations (1) to (9), r0 is the distance between the target and the reference station, r1 is the distance between the target and the receiving station (1), r2 is the distance between the target and the receiving station (2), and r3 is the distance between the target and the receiving station (3).
Further, Ztgt is the target position (vector), and Zsnsi is the position (vector) (i=1, 2, 3) of the receiving station (i).
fi(Ztgt) is a function of the target position Ztgt, the function regarding the distance difference of the receiving station (i).
|A| is the Euclidean norm of a vector A, and |Ztgt−Zsnsi| (i=1, 2, 3) is the Euclidean norm of a vector Ztgt−Zsnsi and shows the distance ri between the target and the receiving station (i).
Further, X, Y, and Z in the right-hand side of the equation (8) are X, Y, and Z components of the target position, and Xsnsi, Ysnsi, and Zsnsi in the right-hand side of the equation (9) are X, Y, and Z components of the receiving station.
In addition, A′ shows the transposition of the vector A, and the same goes for a matrix.
When the equations (1) to (3) are generalized for each of the receiving stations (i) (i=1, 2, 3), the following equation (10) is provided.fi(Ztgt)=ri−r0=|Ztgt−Zsnsi|−|Ztgt−Zsns0|  (10)
When the distance differences f1(Ztgt), f2(Ztgt), and f3(Ztgt) between the receiving stations (1) to (3) and the reference station are differentiated with respect to the target position vector, Jacobians which are the results of the differentiation are given by the following equations (11) to (13).
                              G          1                =                  [                                                                                          ∂                                          f                      1                                                        /                                      ∂                    x                                                                                                                    ∂                                          f                      1                                                        /                                      ∂                    y                                                                                                                    ∂                                          f                      1                                                        /                                      ∂                    z                                                                                ]                                    (        11        )                                          G          2                =                  [                                                                                          ∂                                          f                      2                                                        /                                      ∂                    x                                                                                                                    ∂                                          f                      2                                                        /                                      ∂                    y                                                                                                                    ∂                                          f                      2                                                        /                                      ∂                    z                                                                                ]                                    (        12        )                                          G          3                =                  [                                                                                          ∂                                          f                      3                                                        /                                      ∂                    x                                                                                                                    ∂                                          f                      3                                                        /                                      ∂                    y                                                                                                                    ∂                                          f                      3                                                        /                                      ∂                    z                                                                                ]                                    (        13        )                                G        =                  (                                                                      G                  1                                                                                                      G                  2                                                                                                      G                  3                                                              )                                    (        14        )            
In the equations (11) to (13), G1 is the Jacobian of the distance difference of the receiving station (1), G2 is the Jacobian of the distance difference of the receiving station (2), and G3 is the Jacobian of the distance difference of the receiving station (3).
G is a combination of G1, G2, and G3, and the matrix G is referred to as the all-receiving-stations Jacobian matrix in some cases.
∂fi/∂x (i=1, 2, 3) is the partial differential of the distance difference of each receiving station (i) with respect to x, ∂fi/∂y (i=1, 2, 3) is the partial differential of the distance difference of each receiving station (i) with respect to y, and ∂fi/∂z (i=1, 2, 3) is the partial differential of the distance difference of each receiving station (i) with respect to z.
The all-receiving-stations Jacobian matrix G in the equation (14) increases in size with respect to the row direction of G with increase in the number of receiving stations.
In the example of the equations (11) to (14), the target to be estimated is the target position [x, y, z]. When the target to be estimated includes the target position (x, y, z), a target velocity (vx, vy, vz), and a receiving station clock bias δt, the target to estimated is [x, y, z, vx, vy, vz, δt] and is enlarged with respect to the column direction of G. More specifically, the size of the matrix G changes according to the number of receiving stations for the target to be estimated and the dimensionality of vectors.
When calculating evaluation indices DOPs for the positioning accuracy for the target, a matrix D is calculated from the all-receiving-stations Jacobian matrix G of the target position at one certain point, as shown in the following equation (15).D=inv(G′G)  (15)
In the equation (15), inv( ) means a function of calculating an inverse matrix.
In the example shown in FIG. 16, the matrix D in the equation (15) has 3 rows and 3 columns as shown in the following equation (16).
Further, when up to the three-dimensional position and the receiving station clock bias are included in the target to be estimated, the matrix D has 4 rows and 4 columns as shown in the following equation (17). D11, . . . , and D44 show elements of the matrix.
                    D        =                  (                                                                      D                  11                                                                              D                  12                                                                              D                  13                                                                                                      D                  21                                                                              D                  22                                                                              D                  23                                                                                                      D                  31                                                                              D                  32                                                                              D                  33                                                              )                                    (        16        )                                D        =                  (                                                                      D                  11                                                                              D                  12                                                                              D                  13                                                                              D                  14                                                                                                      D                  21                                                                              D                  22                                                                              D                  23                                                                              D                  24                                                                                                      D                  31                                                                              D                  32                                                                              D                  33                                                                              D                  34                                                                                                      D                  41                                                                              D                  42                                                                              D                  43                                                                              D                  44                                                              )                                    (        17        )            
As the types of evaluation indices DOPs for the positioning accuracy for the target, there are GDOP (Geometric Dilution Of Precision) regarding geometry, PDOP (Position Dilution Of Precision) regarding position, HDOP (Horizontal Dilution Of Precision) regarding horizontal position, VDOP (Vertical Dilution Of Precision) regarding vertical position, and TDOP (Time Dilution Of Precision) regarding clock bias, etc.
Arithmetic expressions for calculating GDOP, PDOP, HDOP, VDOP, and TDOP are expressed by the following equations (18) to (22), respectively.
Hereafter, the matrix D when up to the three-dimensional position and the receiving station clock bias are included in the target to be estimated, as shown in the equation (17), is assumed. When up to the three-dimensional position is included in the target to be estimated, as shown in the equation (16), GDOP=PDOP.GDOP=sqrt(D11+D22+D33+D44)  (18)PDOP=sqrt(D11+D22+D33)  (19)HDOP=sqrt(D11+D22)  (20)VDOP=sqrt(D33)  (21)TDOP=sqrt(D44)  (22)
Although the evaluation indices DOP for the positioning accuracy for the target are calculated the above-mentioned way, the conventional positioning and tracking device has problems which are divided into three general groups.
[Problem 1]
When the number of receiving stations does not reach the one required to estimate the target position (when the distance differences cannot be acquired), there is a problem that the three-dimensional position of the target cannot be calculated and the calculation of DOPs cannot be performed either.
FIG. 17 is an explanatory drawing showing a situation in which the distance cannot be acquired by the receiving station (1) because of blocking by buildings, and the distance difference of the receiving station (1) cannot be acquired as a result.
FIG. 18 is an explanatory drawing showing an example of the problem 1, a number line in an upper row shows an input situation of each receiving station for which the distance difference is acquired, and a number line in a lower row shows a positioning situation.
In the example of FIG. 18, although at a time t1, the distance differences between the reference station and the receiving stations (1) to (3) are acquired, and the calculation of the three-dimensional position is performed, at a time t2, the distance of the receiving station (1) cannot be acquired under the influence of blocking by buildings or the like and the distance difference between the reference station and the receiving station (1) cannot be acquired, and therefore the calculation of the three-dimensional position cannot be performed. At a time t3, the calculation of the three-dimensional position is performed in the same way as that at the time t1.
[Problem 2]
There is a problem that although the number of receiving stations reaches the one required to estimate the target position, the positioning may become impossible from a viewpoint of numerical computations because the placement of each receiving station is bad.
FIG. 19 is an explanatory drawing showing an example of the problem 2, a number line in an upper row shows an input situation of each receiving station for which the distance difference is acquired, and a number line in a lower row shows a positioning situation.
In the example of FIG. 19, although at times t1 and t3, the distance differences between the reference station and the receiving stations (1) to (3) are acquired, and the calculation of the three-dimensional position is performed, at a time t2, the positioning becomes impossible from a viewpoint of numerical computations because the placement of each receiving station is bad.
Although the matrix D calculated by using the equation (16) or (17) is used when calculating the evaluation indices DOP for the positioning accuracy, the row elements of the all-receiving-stations Jacobian matrix G shown in the equation (14) becomes substantially the same as each other because the distances and the distance differences for the target are substantially the same as each other in the case of the placement of each receiving station at the time t2.
As a result, when calculating the inverse matrix of the all-receiving-stations Jacobian matrix G by using the equation (15), there is a case in which the rank of the matrix drops and the inverse matrix cannot be calculated.
Therefore, it is expected that even if the all-receiving-stations Jacobian matrix G can be calculated, the matrix D necessary for the calculation of the evaluation indices DOPs for the positioning accuracy cannot be calculated and therefore the evaluation indices DOPs for the positioning accuracy cannot be calculated, and, as a result, the positioning and tracking device falls into a state in which the positioning and tracking device cannot calculate the three-dimensional position from a viewpoint of numerical computations.
It is also expected that even when the evaluation indices DOPs for the positioning accuracy can be calculated, the values of the DOPs are large. In principle, it is natural that the values of the DOPs are large.
[Problem 3]
In order to improve a situation in which, for example, the three-dimensional position cannot be calculated at the time t2, as shown in Problems 1 and 2, there can be provided a method of performing interpolation calculation or extrapolation calculation by using three-dimensional positions measured at two or more observation points to measure the three-dimensional position at the time t2.
FIG. 20 is an explanatory drawing showing an example of performing interpolation calculation or extrapolation calculation to measure the three-dimensional position at the time t2. The three-dimensional position acquired through interpolation calculation or extrapolation calculation is expresses by ▴.
Although the three-dimensional position can be measured if interpolation calculation or extrapolation calculation is performed even in a case in which the number of receiving stations does not reach the one required to estimate the target position, the evaluation indices DOPs for the positioning accuracy cannot be calculated.
Although the problems arising in the case of measuring the target position from the distance differences between the reference station and the receiving stations (1) to (3) is explained until now, the problems 1 to 3 similarly arise in the case of measuring the target position from the distances between the reference station and the receiving stations (1) to (3), as shown in FIG. 21.